Tensor field

Let $(𝑀, 𝒜)$ be a $𝐶 𝛼$ -manifold. A $𝐶 𝛼$ -tensor field is a generalization of a vector field where we assign a tensor smoothly to every point in $𝑀$ . A homogenous tensor field $𝑋$ of type $(𝑝, 𝑞)$ is a $𝐶 𝛼 (𝑀)$ -multilinear map

𝑋 : Ω 1 𝑀 \times \dots \times Ω 1 𝑀 ⏟___⏟___⏟ 𝑝 \times 𝔛 (𝑀) \times \dots \times 𝔛 (𝑀) ⏟____⏟____⏟ 𝑞 \to 𝐶 𝛼 (𝑀)

where $Ω 1 𝑀$ and $𝔛 (𝑀)$ denote the spaces of 1-forms and vector fields respectively. The $𝐶 𝛼 (𝑀)$ -module of all such tensor fields is denoted $T 𝑝 𝑞 𝑀$ . A general nonhomogenous tensor field is a direct sum of tensor fields.

As a section

The above definition is equivalent to a $𝐶 𝛼$ -section of the tensor product of $𝑝$ copies of the tangent bundle and $𝑞$ copies of the cotangent bundle

𝑇 𝑝 𝑞 𝑀 = (𝑇 𝑀) \otimes 𝑝 \otimes (𝑇 * 𝑀) \otimes 𝑞 𝑋 \in T 𝑝 𝑞 𝑀 = Γ 𝛼 (𝑀, 𝑇 𝑝 𝑞 𝑀)

and a general (non-homogenous) tensor field is a $𝐶 𝛼$ -section of a sum bundle.

Proof

#missing/proof

Further terminology

A totally contravariant, totally antisymmetric tensor field is a Differential form.
Often physicists view the Tensor field transformation laws as the definition of a tensor field.
A metric tensor enables Raising and lowering of indices.

Local coördinates

Let $𝑥 : 𝑈 ↪ ℝ 𝑚$ be a chart. Restricted to $𝑈$ , we may write a smooth tensor field $𝑋 \in T 𝑝 𝑞 𝑀$ in the form

𝑋 = 𝑋 𝜇 1 \dots 𝜇 𝑝 𝜈 1 \dots 𝜈 𝑝 𝜕 𝜇 1 \otimes \dots \otimes 𝜕 𝜇 𝑝 \otimes d 𝑥 𝜈 1 \otimes \dots \otimes d 𝑥 𝜈 𝑞

#state/tidy | #lang/en | #SemBr